Effect of Cattaneo-Christov approximation for viscoelastic fluid with carbon nanotubes on flow and heat transfer

The current work studies the motion of viscoelastic liquid saturated with carbon nanotubes over a stretching surface in a Darcy porous medium analytically below an influence of Cattaneo-Christov heat flux. The carbon nanotubes (CNTs) act as nanoparticles which are then appended into the base fluid. Water and kerosene are used as a base fluid with two types of CNTs, namely, Single-wall carbon nanotubes and Multiwall carbon nanotubes. Carbon nanotubes possess a wide range of industrial and biomedical applications including energy production, nuclear reactor cooling, and galaxy cooling applications because they can expand the thermal and mechanical properties of base things. As a result, the carbon nanotubes used in the mentioned fields are being investigated for their potential in heat transfer applications. Governing equations formulated using the Partial differential equations have converted to Ordinary differential equations exhausting the appropriate comparison transformation process. An influence of some relevant constraints on velocity and temperature is evaluated in details. The Cattaneo-Christov heat transfer model is utilized to investigate the heat transfer individualities with varying thermal conductivity consuming the attributes of the Appell hypergeometric function. The impacts of the emerging parameters on the profiles are depicted through graphical representations and analytically constructed tables. Considering its usefulness in modulating temperature distribution in different industrial application, including solar collector design, electronic cooling, building ventilation, etc. According to our findings, the temperature profile exhibits an enhancement with the thermal radiation parameter and the viscous-elastic fluids. In addition, when compared to the classical Fourier's law of heat conduction, the temperature profile and thermal boundary layer thickness for the Cattaneo-Christov heat flux model are lower.


Physical model and solution
An investigation of incompressible flow of second grade liquid/Walters' B liquid created by a continuously stretched sheet under an influence of inclined magnetic field and Darcy porous medium was performed (see Fig. 1). In two instances of fluids, the governing two-dimensional boundary layer flows are examined.
Governing equation can be formulated as follows where k * = −α ρ f is the elastic parameter, k * > 0 is for viscoelastic fluid 41 , k * < 0 is for the second grade fluid 42 , while k * = 0 is for the Newtonian fluid 6 . Using the Cattaneo-Christov model 24

one can find
Here k is the thermal conductivity and λ is the heat flow relaxation time. Equation (4) can be simplified to the standard Fourier's equation of heat transfer when λ = 0.
When q is included in Eqs. (3) and (4), the temperature profile can be attained as follows The following B. Cs are used Jafarimoghaddam et al. 40 .
Blasius similarity transformation is given by    (2) and (5) The used B Cs is is the thermal diffusivity, and The nanofluids constants are mathematically defined as: (see Muhammad et al. [43][44][45][46][47] Tables 1 and 2 show the thermal properties of different nanoparticles and base fluids. The necessary solution is expected to be around the form, based on the exact analytical model for Eq. (8) and the accompanying boundary condition (10) and the preceding choice of f η is The following equation, which is produced by utilizing Eq. (11) in Eq. (8), allows to determine the unknown  It is clear that in the case of A > 0, the thermal problem solution deviates from its correct physical meaning; as a result A < 0 remains to solve the possible solution. Therefore, The recently discovered threshold condition for the presence of a thermal solution become If the circumstances f ηη (0) 2 > γ is available Validation study. Here, we note that, . The following Table 3 shows related works by other authors and finding existing results.

Results and discussion
The investigation is simplified in addition by the scoping review of the velocity and temperature equations, which results in a set of ODEs. Exact analytical solutions for momentum and temperature profiles can be achieved by using the appropriate similarity variable. As a result of the multiple graphs presented above that build on the subject, we now understand the technology involved in such fascinating dynamics. Furthermore, the numbers of similar visuals provide a comparative of the transverse, axial, and temperature profiles of SWCNT and MWCNT with the solid volume fraction fixed, with dashed lines reflecting SWCNTs and solid lines indicating MWCNTs. Figure 2 demonstrates the relaxation time versus viscoelastic parameter for varying magnetic parameter M. The difference is essential because it gives acute curves and limitations for the thermal explanations that will happen after the overall non-Fourier temperature profile is implanted. The magnetic constraint increases the extent of the boundary layer increases in the stretching surface.

Amin et al. 2021
Non-Newtonian www.nature.com/scientificreports/ Figure 3 signifies the depiction of Darcy model Da -1 on f (η) while the stretching sheet is more than zero. The f (η) of MWCNT is more than that of SWCNT. The f (η) enhances as the Darcy number also increases. The magnetic field's inclined parameter is 90 degree. Figure 4 portrays the various values of magnetic parameter M on f (η) with deference to the similarity variable η at the point when the stretching boundary layer is more than nothing. As seen in the diagram, the Lorentz effect is reduced when the raising the M decrease the thickness of the boundary layer, resulting in increased shear stress on the wall. The reason for this is because when the magnetic parameter rise, the boundary layer increases, which is accompanied by a increase in the velocity gradient. In addition, the volume fraction is enhances, the boundary layer thickness enhances. Figure 5 portrays the various values of viscoelastic k 1 on transverse velocity f (η) regarding the similarity variable η at the point when the stretching boundary layer is bigger than nothing. The viscoelastic constraint raises the extent of the boundary layer enhances in the stretching surface. The transverse velocity which occurs where  www.nature.com/scientificreports/ there is porous stretching/shrinking sheet. In the presence of stronger viscoelastic fluid higher, will increase transverse velocity that would show off in higher altitudes, namely η ≥ 1. The reason is behind complex rheological behavior of Walter´s liquid B which shows more strength where there is more velocity. At higher altitude where η ≥ 1, the third term in 2f η f ηηη − ff ηηηη − f 2 ηη representing shear stress inside fluid layers gradually vanishes and hence cause whole the term to grow. Consequently, it is expected that at higher altitude, effect of viscoelastic fluid would be bolder. Figures 6, 7 and 8 demonstrate the behaviour of the f η (η) on the similarity variable η for altered values of Darcy number and magnetic constraint. Figure 6 show that when the magnetic rises, the boundary layer thickness also decreases. In both cases of stretching/shrinking the surface, raising the magnetic parameter increases the axial velocity in the flow field. The influence of magnetic field on velocity at wall for the stretched sheet is seen in this diagram, with velocity and overall axial velocity decreasing as the magnetic field magnitude increases.  www.nature.com/scientificreports/ Magnetic field, as previously stated, is an attractive body force whose projection on the x-axis is in the negative x-direction. It indicates that a larger magnetic field value will cause more axial velocity obstruction and, as a result, will diminish it. The imposed boundary condition and magnetic field, on the other hand, are both in the same direction for shrinking sheet. Consequently, the higher magnetic parameter (as a representative of magnetic field) leads to a higher axial velocity. The similar effect is observed at Fig. 7, it means that f η (η) decreases with increasing the Darcy model. The similar effect is observed at Fig. 8, it means that f η (η) decreases with increasing the viscoelastic parameter. As can be observed, the viscoelastic influence on axial velocity varies depending on whether the object is shrinking or stretching. In reality, while stretching a sheet, the force and influence of the boundary condition is more important than the fluid's rheology. An increase in the size of the viscoelastic parameter leads in a comparable rise in the axial velocity profiles in both porous stretching and shrinking instances.  www.nature.com/scientificreports/ The boundary layer thins as a result of non-Newtonian viscoelastic shear stress. Furthermore, the boundary layer is smaller when suction is utilized instead of injection. Figures 9, 10 and 11 represent the temperature fields θ(η) on the similarity variableη, at the point when the stretching limit is more noteworthy than nothing. In Fig. 9, the magnetic parameter increases, the temperature profile also increases. Figure 10 shows different values of radiation. When θ(η) is enhanced, the radiation parameter also rises. Consequently, thermal radiation improves the nanofluids thermal diffusivity, i.e., for emergent values of radiation parameter N R , heat will be supplemented to the regime and temperatures improved accordingly. As mentioned for heat transfer of flows over a stretching sheet, fluid temperature higher than both the wall temperature and the ambient temperature near the wall is physically achievable. Here we discuss of forced flow over a stretching sheet, we now look at heat transport in the presence of radiation. The effect of heat conductivity is amplified by the radiation. Radiation has the effect of dampening or enhancing heat transmission in a linear  www.nature.com/scientificreports/ manner. In Fig. 11, the viscoelastic parameter increases, the temperature profile also increases. Because the shear rate is higher near the solid wall, the impact of the viscosity parameter, which can be considered as the same of fluid viscosity, on temperature profiles is more obvious. It takes longer for fluid molecules to transfer energy to neighbouring molecules. Therefore, the temperature curves demonstrate a increasing nature/behaviour. Figure 12 illustrates the flow patterns for different parameters. In this section, we'll look at how to simplify techniques in the streamline circumstances of stretching cases. Highlights the pattern of streamlines for stretching boundary for various values of magnetic parameter with fixed parameters k 1 = Da −1 = 1, φ = 0.1 . The flow field is regularized when the magnetic field occurs at particular subsequent places, as predicted by the physical theory. On the other hand, results in the removal of the streamline in the circular configuration.  www.nature.com/scientificreports/

Conclusions
The Cattaneo-Christov heat flux is studied in carbon nanotubes with porosity in boundary layer flows of two viscoelastic fluids across a stretching surface. In many other terms, the parameters as with inclined magnetic field, viscoelastic constraint and Prandtl number strictly reduce the relaxation time coefficient. The analytical explanation of heat transfer presences in term of the Appell hypergeometric function. The temperature distribution inside the stretching/shrinking sheet is also controlled by the Prandtl number, thermal radiation, mass transpiration, heat source/sink, and magnetic parameter variables.
• Axial and transverse velocities increase as the parameters such as magnetic parameter, Darcy number increase in both SWCNTS and MWCNTs cases. • The effect of the thermal radiation parameter raises the temperature enhances it.
• When the value of the viscoelastic parameter is enhanced, the fluid temperature increases in both stretching/ shrinking conditions. • The results of this research are very similar to those of Jafarimoghaddam et al. 40